Numerical Approximation of Partial Differential Equations Involving Fractional Differential Operators

Abstract

The negative powers of an elliptic operator can be approximated via its Dunford-Taylor integral representation, i.e. we approximate the Dunford-Taylor integral with an exponential convergent sinc quadrature scheme and discretize the integrand (a diffusion-reaction problem) at each quadrature point using the finite element method. In this work, we apply this discretization strategy for a parabolic problem involving fractional powers of elliptic operators and a stationary problem involving the integral fractional Laplacian. The approximation of the parabolic problem is twofold: the homogenous problem and the non-homogeneous problem. We propose an approximation scheme for the homogeneous problem based on a complex-valued integral representation of the solution operator. An exponential convergent sinc quadrature scheme with a hyperbolic contour and a complex-valued finite element method are developed. The approximation of the non-homogeneous problem in space follows the same idea from the homogeneous problem but we need to additionally discretize the problem in the time domain. Here we consider two different approaches: a pseudo-midpoint quadrature scheme in time based on Duhamel’s principle and the Crank-Nicolson time stepping method. Both methods guarantee second order convergence in time but require different sinc quadrature schemes to approximate the corresponding fractional operators. The time stepping method is stable provided that the sinc quadrature spacing is sufficiently small. In terms of the approximation of the stationary problem involving integral fractional Laplacian, we consider a Dunford-Taylor integral representation of the bilinear form in the weak formulation. After approximating the integral with a sinc quadrature scheme, we need to approximate the integrand at each quadrature point which contains a solution of a diffusion-reaction equation defined on the whole space. We approximate the integrand problem on a truncated domain together with the finite element method.

For both problems, we provide L² error estimates between solutions and their final approximations. Numerical implementation and results illustrating the behavior of the algorithms are also provided.